“...where it’s at.” - Is that grammatical?

Yes, your statement was perfectly grammatical.

Your customer is correct that you don’t have to say ‘at’: “where it is” is every bit as acceptable as—and in some circumstances more acceptable than—“where it’s at”, but “where it’s at” is not ungrammatical as such.

Where is usually (or perhaps I should rather say traditionally) labelled a relative adverb in this usage. It’s not really an adverb in the narrower sense of the verb (it doesn’t directly modify the verb in a clause); it is in many ways more akin to a relative pronoun, especially in that it always has an antecedent, whether explicit or not. This antecedent is essentially a prepositional phrase of some kind, with the object of the preposition being the place that functions as the ‘actual’ antecedent; if there is no explicit place in the main clause, we assume a generic antecedent like “place” or something like that.

In other words, you can usually replace where in a relative clause with a prepositional phrase: “in which [place]”, “at which [place]”, “to which [place]”, etc. Or, since we’re talking about relative clauses here, “that/what … in/at/to/etc.”. Compare:

This is the school where my kids go // This is the school that my kids go to.
This is the house where/in which I live // This is the house that I live in.
That is the college where/at which I trained // That is the college that I trained at.

In all these cases, where by itself (without the preposition) carries the meaning of the entire prepositional phrase.

Since we always have the corresponding ‘full’ phrase somewhere in the back of our heads, however, and know that underlyingly we’re talking about ‘going to school’, ‘living in a house’, and ‘training at college’, we sometimes add the preposition anyway. Essentially, the only thing this changes is that where then does not stand in for the entire prepositional phrase, but only for the place—it becomes entirely synonymous with that.

In some cases, this is actually necessarily if you want to specify whether you’re talking about direction or location. A taxi driver, for instance, might ask you, “Where to?” to ask you where you want him to take you, but he wouldn’t just ask “Where?”, because that might as well be asking you where you are or half a dozen other things.

In most cases, though, adding the preposition is unnecessary—and as with many things in language that are not necessary, some people have interpreted its lack of necessity as evidence that it is in fact wrong. Hence, we end up with people making outdated claims that sentences like “Where are you going to?” are ‘not grammatically correct’—which is nonsense, of course.

The preposition at is an interesting case in this connection. In general, adding the preposition is associated with spoken language and a somewhat colloquial register. But when the verb is the copula (be) and the preposition is at, there is the additional circumstance that the phrasing “where X is at” has somehow become extremely widespread (even in contexts where you would normally use in as the preposition) especially in AAVE, being used in many cases where most dialects would simply have “where X is”.

This association has done a lot to stigmatise the collocation of where and at, since AAVE is one of the dialects traditionally most severely looked down upon as uncultured and uneducated.

Going off into wild speculation, I would guess that is why your customer reacted to your use of “where it’s at”, even though in all likelihood he would not have batted an eyelid at “I can’t find the box where it’s in right now” or other similar examples.